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Abstract

Most existing methods for solving two-phase flow problems in porous media do not take the physically feasible saturation fractions between 0 and 1 into account, which often destroys the numerical accuracy and physical interpretability of the simulation. To calculate the solution without the loss of this basic requirement, we introduce a variational inequality formulation of the saturation equilibrium with a box inequality constraint, and use a conservative finite element method for the spatial discretization and a backward differentiation formula with adaptive time stepping for the temporal integration. The resulting variational inequality system at each time step is solved by using a semismooth Newton algorithm. To accelerate the Newton convergence and improve the robustness, we employ a family of adaptive nonlinear elimination methods as a nonlinear preconditioner. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm. A comparison is also included to show the superiority of the proposed fully implicit approach over the classical IMplicit Pressure-Explicit Saturation (IMPES) method in terms of the time step size and the total execution time measured on a parallel computer.

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The authors would like to express their appreciation to the anonymous reviewers for their invaluable comments, which have greatly improved the quality of the paper. The work was supported in part by Special Project on High-Performance Computing under the National Key R&D Program (2016YFB0200603) and National Natural Science Foundation of China (11571100, 91530323, 11272352). S. Sun was also supported by KAUST through the grant BAS/1/1351-01-01. C. Yang was also supported by Key Research Program of Frontier Sciences from CAS through the grant QYZDB-SSW-SYS006.

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